An [[n, k]] quantum stabiliser code Q is said to be a best known [[n, k]] quantum code (BKQC) if C has the highest minimum weight among all known [[n, k]] quantum codes. The acronym QECC (Quantum Error Correcting Code) will be used to more easily distinguish from the best known linear codes database (BKLC).
Magma currently has a database for binary quantum codes, though it should be noted that these codes are considered to be over the alphabet GF(4), not GF(2). The database for codes over GF(4) currently contains constructions of all best known quantum codes of length 35. This includes self-dual quantum codes up to length 35, which are stored in the database as dimension 0 quantum codes.
Quantum codes of length up to 12 are optimal, in the sense that their minimum weights meet the upper bound. Thus the user has access to 665 best-known binary quantum codes.
The Magma QECC database uses the tables of bounds and constructions compiled by Markus Grassl (Karlsruhe), available online at [Gra], which are based on the results in [CRSS98]. Good codes have also been contributed by Eric Rains and Zlatko Varbanov.
The user can display the method used to construct a particular QECC code through use of a verbose mode, triggered by the verbose flag BestCode. When it is set to true, all of the functions in this section will output the steps involved in each code they construct.
Given a finite field F, and positive integers n and k such that k ≤n, return an [[n, k]] quantum code over F which has the largest minimum weight among all known [[n, k]] quantum codes. A second boolean return value signals whether or not the desired code exists in the database.The database currently exists for GF(4) (which are in fact binary quantum codes) up to length 35.
> F<w> := GF(4); > Q := QECC(F,25,16); > Q:Minimal; [[25, 16, 3]] Quantum code over GF(2^2) > time WD_S, WD_N, WD := WeightDistribution(Q); Time: 0.010 > WD_S; [ <0, 1>, <1, 2>, <2, 1>, <14, 4>, <15, 16>, <16, 38>, <17, 79>, <18, 126>, <19, 129>, <20, 77>, <21, 27>, <22, 9>, <23, 3> ] > WD_N; [ <0, 1>, <1, 2>, <2, 1>, <3, 399>, <4, 6527>, <5, 75363>, <6, 707543>, <7, 5404369>, <8, 34084490>, <9, 180107319>, <10, 804255370>, <11, 3052443894>, <12, 9883860222>, <13, 27348684334>, <14, 64649758926>, <15, 130286413858>, <16, 222912028997>, <17, 321704696752>, <18, 387985433701>, <19, 385943417035>, <20, 310898936275>, <21, 197566276671>, <22, 95232787563>, <23, 32688613821>, <24, 7109768160>, <25, 735493959> ] > WD; [ <3, 399>, <4, 6527>, <5, 75363>, <6, 707543>, <7, 5404369>, <8, 34084490>, <9, 180107319>, <10, 804255370>, <11, 3052443894>, <12, 9883860222>, <13, 27348684334>, <14, 64649758922>, <15, 130286413842>, <16, 222912028959>, <17, 321704696673>, <18, 387985433575>, <19, 385943416906>, <20, 310898936198>, <21, 197566276644>, <22, 95232787554>, <23, 32688613818>, <24, 7109768160>, <25, 735493959> ]So the [[25, 16]] code is impure, and has a minimum distance of 3.
> F<w> := GF(4); > C := QECC(GF(4),8, 0); > C; [[8, 0, 4]] self-dual Quantum code over GF(2^2), stabilised by: [ 1 0 0 1 0 1 1 0] [ w 0 0 w 0 w w 0] [ 0 1 0 1 0 1 0 1] [ 0 w 0 w 0 w 0 w] [ 0 0 1 1 0 0 1 1] [ 0 0 w w 0 0 w w] [ 0 0 0 0 1 1 1 1] [ 0 0 0 0 w w w w]
> SetVerbose("BestCode",true);
> F<w> := GF(4);
> Q := QECC(F,25,11);
Construction of a [[ 25 , 11 , 4 ]] Quantum Code:
[1]: [[40, 30, 4]] Quantum code over GF(2^2)
QuasiCyclicCode of length 40 stacked to height 2 with generating
polynomials: 1, w^2*x^4 + w*x^3 + w^2*x^2 + w*x + w^2, x^4 + w^2*x^2 +
w^2*x + w^2, x^4 + w*x^3 + x^2, w*x^4 + x^3 + w^2*x, w*x^4 + x^3 +
x^2 + x, w^2*x^4 + x^2 + w*x, x^4 + w^2*x^3 + w^2*x^2 + x + 1, w,
x^4 + w^2*x^3 + x^2 + w^2*x + 1, w*x^4 + x^2 + x + 1, w*x^4 + w^2*x^3
+ w*x^2, w^2*x^4 + w*x^3 + x, w^2*x^4 + w*x^3 + w*x^2 + w*x, x^4 +
w*x^2 + w^2*x, w*x^4 + x^3 + x^2 + w*x + w
[2]: [[21, 11, 4]] Quantum code over GF(2^2)
Shortening of [1] at { 2, 3, 4, 6, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19,
21, 24, 28, 34, 37 }
[3]: [[25, 11, 4]] Quantum code over GF(2^2)
ExtendCode [2] by 4
> Q:Minimal;
[[25, 11, 4]] Quantum code over GF(2^2)